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Electron momentum distribution in polycrystalline iron
Solid State Communications,
Vol. 9, pp. 457—459, 1971.
Pergamon Press.
Printed in Great Britain
ELECTRON MOMENTUM DIST~BUTIONIN POLYCRYSTALLINE IRON
J.
Feisteiner, R. Fox and S. Kahane
Department of Physics, Technion
—
Israel Institute of Technology, Haifa, Israel
(Received 21 December 1970 by E. Bursiein)
The Cornpton profile of polycrystalline iron has been measured using 59.54 keV WArn gamma rays with particular emphasis on the high-momentum region. The data are found to be in good agreement with calculations based on Hartree—Fock wave functions, in marked contrast with recent MoKaX-ray measurements on titanium. It is found that seven 3d electrons in iron fit best the experimental data, in agreement with Mässbauer isomer shift measurements.
THE USE of Compton scattering of X-rays for the purpose of studying the electron momentum density in solids was begun a number of years ago by DuMond.’ This technique is being continued today by vario~jsgroups.25 In particular the first measurements on transition element, titanium, was performed recently,5 and a very large discrepancy was found between the measured Compton profile and that calculated from Hartree—Fock wave functions for momenta greater than approximately three times the Fermi momentum (see Fig. 1 in reference 5). This result is surprising since the high-momentum tail is primarily due to the inner bound electrons which are expected to be changed little upon going from the free atom to the metallic state.
The i7keVMoKaX-rays that were used in the above investigation have wavelengths of the order of the interelectron spacing thus raising doubt as to their sensitivity to the individual electron momenta. A much cleaner method for studying the individual electron momentum density is the Coinpton scattering of photons of shorter wavelength. For example, 60 keV gamma rays have been recently used to study the electron momentum distributions in graphite6 and aluminum.7 457
In the present investigation the Compton profile of polycrystalline iron was measured using 59.54 keV gamma rays from a 300 mCi 241Am source. ~1~escattered gamma rays were detected at an angle of 165°by a Ge(Li) detector having a resolution of 400 eV FWHM at 60 keV. The detector output was coupled to a 4096-channel multichannel analyzer. The high-energy side of the experimental Cornpton profile, corrected for the finite instrumental resolution and convertedto electron momentum scale, is shown in Fig. 1. Also shown is the Compton profile C of a core of electrons 1s22s22pG3s 3jt calculated from Hartree—Fock atomic wave functions as tabulated by Herman and Skiliman? If the is2 electrons were not excited we would have a core of electrons 2s22p~3s23p6 whose Coinpton profile ~ is also shown in Fig. 1. The areas of the curves C and C are normalized to 18/26 and 16/24 of the experimental curve respectively. The correct theoretical core also shown in Fig. 1, is C above the is2 threshold at 22.8 keV/c and ~ below threshold, appropriately normalized. A discontinuity is observed in the derivative of the Compton profile and is designated by an arrow in Fig. 1. This implies a discontinuity in the electron momentum distribution associated with the Fermi surface. Above this discontinuity a best fit to the experimental points is obtained with C plus the Hartree—Fock contribution of seven 3d electrons, which is in agreement with ~,
458
ELECTRON MOMENTUM DISTRIBUTION IN POLYCRYSTALLINE IRON
Vol. 9, No. 8
Fe
z (key/c)
~.cf,.
FIG. 1. The experimental Compton profile of polycrystalline iron (solid curve). The statistical error is shown near the peak. The curve marked C is calculated from 6core. Hartree—Fock waverepresents functions The C curve for a 1#2s22p~s23p 2s22p63s23p6 core. C is a curve normalized so as to reprerent C above threshold and ~ below threshold. The curves 3d6and 3d7are the superposition of C with the Hartree—Fock contributions of six and seven 3d electrons respectively. M6ssbat~risomer shift measurements.9 This theoretical curve, (3d7), is shown in Fig. 1 along with the free atom curve of six 3d electrons (3d6). The good agreement of the experimental points with the theoretical curve C is in marked contrast with the very large disagreement found in the results of reference 5 on another transition element, titanium, It is of interest to examine how sensitive our results are to the particular wave functions
—.
45
~
z (k.V/~)
FIG. 2. The high-momentum region of the Compton profile in an expanded scale. The curyes C( ), CC )~and C ( ) are based on the H!rrnan—Skillman wave functions. The bottom curve C (— -) is based on Clementi wave functions. The experimental points are shown in their statistical errors. —
—
—
used. We thus compare the Herman—Skiiman profile for ~ with that obtained from reference 10 based on Clementi wave functions11 for iron. The curves C, ~, C based on the Herman—Skiliman wave functions and ~ based on the Clementi wave functions are shown in Fig. 2 in an expanded scale for the high-momentum region from 35 keV/c to 5OkeV/c. Although the two ~ curves are in fair agreen ant, the Clementi profile has a more restricted tail.
REFERENCES 1.
DUMOND J.W.M., Phys. Rev. 33, 643 (1929); Rev, mod. Phys. 5, 1(1933).
2.
THEODOSSIOU A. and VOSNIDIS P., Phys. Rev. 145, 458 (1966).
3.
COOPER M. and WILLIAMS B.G., Phil. Mag. 17, 1079 (1968).
4.
PHILLIPS W.C. and WEISS R.J., Phys. Rev. 171, 790 (1968).
5. 6.
WEISS R.J., Phys. Rev. Leu. 24, 883 (1970). FELSTEINER J., FOX R. and KAHANE S., Phys. Lett. A 33A, 442 (1970).
7.
FELSTEINER J., FOX R. and KAHANE S., Solid State Commun. 9, 61(1971).
8.
HERMAN F. and SKILLMAN S., Atomic Structure Calculations, Prentice Hall, Englewood Cliffs, New Jersey (1963). 9. WALKER L.R., WERTHEIM G.K. and JACCARINO V., Phys. Rev. Len. 6, 98 (1961). 10. WEISS R.J., HARVEY A. and PHILLIPS W.C., Phil. Mag. 17, 241 (1968). 11.
CLEMENT! E., IBM J. Res. Dev. 9, 2 (1965).
Vol. 9, No. 8
ELECTRON MOMENTUM DISTRIBUTION IN POLYCRYSTALLINE IRON Le profil Compton du fer 241Am polycrystallin a été particulièrement mesuré avec des la rayons en accentuant gammas de grandes 59.54 keV du region des quantitCs de mouvement~Les résultats sont en bon accord avec les calculs au moyen des fonctions d’ondes de Hartree—Fock, en nette opposition avec les mesures récentes par rayons X du MoKa sur le titanium. Le meilleur accord avec les résultats expérimentaux dans Ic fer a été obtenu avec sept electrons 3d, en accord avec les mesures de dCcalage isomérique Mössbauer.
459
Vol. 9, pp. 457—459, 1971.
Pergamon Press.
Printed in Great Britain
ELECTRON MOMENTUM DIST~BUTIONIN POLYCRYSTALLINE IRON
J.
Feisteiner, R. Fox and S. Kahane
Department of Physics, Technion
—
Israel Institute of Technology, Haifa, Israel
(Received 21 December 1970 by E. Bursiein)
The Cornpton profile of polycrystalline iron has been measured using 59.54 keV WArn gamma rays with particular emphasis on the high-momentum region. The data are found to be in good agreement with calculations based on Hartree—Fock wave functions, in marked contrast with recent MoKaX-ray measurements on titanium. It is found that seven 3d electrons in iron fit best the experimental data, in agreement with Mässbauer isomer shift measurements.
THE USE of Compton scattering of X-rays for the purpose of studying the electron momentum density in solids was begun a number of years ago by DuMond.’ This technique is being continued today by vario~jsgroups.25 In particular the first measurements on transition element, titanium, was performed recently,5 and a very large discrepancy was found between the measured Compton profile and that calculated from Hartree—Fock wave functions for momenta greater than approximately three times the Fermi momentum (see Fig. 1 in reference 5). This result is surprising since the high-momentum tail is primarily due to the inner bound electrons which are expected to be changed little upon going from the free atom to the metallic state.
The i7keVMoKaX-rays that were used in the above investigation have wavelengths of the order of the interelectron spacing thus raising doubt as to their sensitivity to the individual electron momenta. A much cleaner method for studying the individual electron momentum density is the Coinpton scattering of photons of shorter wavelength. For example, 60 keV gamma rays have been recently used to study the electron momentum distributions in graphite6 and aluminum.7 457
In the present investigation the Compton profile of polycrystalline iron was measured using 59.54 keV gamma rays from a 300 mCi 241Am source. ~1~escattered gamma rays were detected at an angle of 165°by a Ge(Li) detector having a resolution of 400 eV FWHM at 60 keV. The detector output was coupled to a 4096-channel multichannel analyzer. The high-energy side of the experimental Cornpton profile, corrected for the finite instrumental resolution and convertedto electron momentum scale, is shown in Fig. 1. Also shown is the Compton profile C of a core of electrons 1s22s22pG3s 3jt calculated from Hartree—Fock atomic wave functions as tabulated by Herman and Skiliman? If the is2 electrons were not excited we would have a core of electrons 2s22p~3s23p6 whose Coinpton profile ~ is also shown in Fig. 1. The areas of the curves C and C are normalized to 18/26 and 16/24 of the experimental curve respectively. The correct theoretical core also shown in Fig. 1, is C above the is2 threshold at 22.8 keV/c and ~ below threshold, appropriately normalized. A discontinuity is observed in the derivative of the Compton profile and is designated by an arrow in Fig. 1. This implies a discontinuity in the electron momentum distribution associated with the Fermi surface. Above this discontinuity a best fit to the experimental points is obtained with C plus the Hartree—Fock contribution of seven 3d electrons, which is in agreement with ~,
458
ELECTRON MOMENTUM DISTRIBUTION IN POLYCRYSTALLINE IRON
Vol. 9, No. 8
Fe
z (key/c)
~.cf,.
FIG. 1. The experimental Compton profile of polycrystalline iron (solid curve). The statistical error is shown near the peak. The curve marked C is calculated from 6core. Hartree—Fock waverepresents functions The C curve for a 1#2s22p~s23p 2s22p63s23p6 core. C is a curve normalized so as to reprerent C above threshold and ~ below threshold. The curves 3d6and 3d7are the superposition of C with the Hartree—Fock contributions of six and seven 3d electrons respectively. M6ssbat~risomer shift measurements.9 This theoretical curve, (3d7), is shown in Fig. 1 along with the free atom curve of six 3d electrons (3d6). The good agreement of the experimental points with the theoretical curve C is in marked contrast with the very large disagreement found in the results of reference 5 on another transition element, titanium, It is of interest to examine how sensitive our results are to the particular wave functions
—.
45
~
z (k.V/~)
FIG. 2. The high-momentum region of the Compton profile in an expanded scale. The curyes C( ), CC )~and C ( ) are based on the H!rrnan—Skillman wave functions. The bottom curve C (— -) is based on Clementi wave functions. The experimental points are shown in their statistical errors. —
—
—
used. We thus compare the Herman—Skiiman profile for ~ with that obtained from reference 10 based on Clementi wave functions11 for iron. The curves C, ~, C based on the Herman—Skiliman wave functions and ~ based on the Clementi wave functions are shown in Fig. 2 in an expanded scale for the high-momentum region from 35 keV/c to 5OkeV/c. Although the two ~ curves are in fair agreen ant, the Clementi profile has a more restricted tail.
REFERENCES 1.
DUMOND J.W.M., Phys. Rev. 33, 643 (1929); Rev, mod. Phys. 5, 1(1933).
2.
THEODOSSIOU A. and VOSNIDIS P., Phys. Rev. 145, 458 (1966).
3.
COOPER M. and WILLIAMS B.G., Phil. Mag. 17, 1079 (1968).
4.
PHILLIPS W.C. and WEISS R.J., Phys. Rev. 171, 790 (1968).
5. 6.
WEISS R.J., Phys. Rev. Leu. 24, 883 (1970). FELSTEINER J., FOX R. and KAHANE S., Phys. Lett. A 33A, 442 (1970).
7.
FELSTEINER J., FOX R. and KAHANE S., Solid State Commun. 9, 61(1971).
8.
HERMAN F. and SKILLMAN S., Atomic Structure Calculations, Prentice Hall, Englewood Cliffs, New Jersey (1963). 9. WALKER L.R., WERTHEIM G.K. and JACCARINO V., Phys. Rev. Len. 6, 98 (1961). 10. WEISS R.J., HARVEY A. and PHILLIPS W.C., Phil. Mag. 17, 241 (1968). 11.
CLEMENT! E., IBM J. Res. Dev. 9, 2 (1965).
Vol. 9, No. 8
ELECTRON MOMENTUM DISTRIBUTION IN POLYCRYSTALLINE IRON Le profil Compton du fer 241Am polycrystallin a été particulièrement mesuré avec des la rayons en accentuant gammas de grandes 59.54 keV du region des quantitCs de mouvement~Les résultats sont en bon accord avec les calculs au moyen des fonctions d’ondes de Hartree—Fock, en nette opposition avec les mesures récentes par rayons X du MoKa sur le titanium. Le meilleur accord avec les résultats expérimentaux dans Ic fer a été obtenu avec sept electrons 3d, en accord avec les mesures de dCcalage isomérique Mössbauer.
459